This is a pretty easy one to solve since the number of votes is not privileged information. Simply publicly record the number of votes as each person goes into vote. This can be verified/observed by whoever wishes to do so. You could even require a large public display at each station. Pair that with a camera record that can also be verified. One person enters, the display should go up by one. And from there you have simple math:
The sums of the polling stations should equal the district.
The sums of the districts should equal the state.
The sums of the states should equal the nation.
I have to say it also makes me wonder what in the world darpa is thinking. Modern technology possible, somewhat trivially, elections that would be nearly impossible to fake in any way, shape, or form. And I'm certain this isn't exactly a secret to darpa.